International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 2 (2017), 136-143 http://www.etamaths.com EXISTENCE OF POSITIVE PERIODIC SOLUTIONS FOR A THIRD-ORDER DELAY DIFFERENTIAL EQUATION FARID NOUIOUA1,2, ABDELOUAHEB ARDJOUNI2,3,∗, ABDELKERIM MERZOUGUI1 AND AHCENE DJOUDI3 Abstract. In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = f (t,x (t) ,x(t − τ(t))) + c(t)x′(t − τ(t)), is considered. By employing Green’s function and Krasnoselskii’s fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order delay differential equation. 1. Introduction Delay differential equations have received increasing attention during recent years since these equa- tions have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering, see the monograph [8, 19] and the papers [1]- [18], [20]- [22], [24]- [27] and the references therein. The second order nonlinear delay differential equation with periodic coefficients x′′ (t) + p (t) x′ (t) + q (t) x (t) = r (t) x′ (t− τ (t)) + f (t,x (t) ,x (t− τ (t))) , has been investigated in [25]. By using Krasnoselskii’s fixed point theorem and the contraction mapping principle, Wang, Lian and Ge obtained existence and uniqueness of periodic solutions. In [22], Ren, Siegmund and Chen discussed the existence of positive periodic solutions for the third-order differential equation x′′′ (t) + p (t) x′′ (t) + q (t) x′ (t) + c (t) x (t) = g (t,x (t)) . By employing the fixed point index, the authors obtained existence results for positive periodic solu- tions. Inspired and motivated by the works mentioned above and the papers [1]- [18], [20]- [22], [24]- [27] and the references therein, we concentrate on the existence of positive periodic solutions for the third-order nonlinear delay differential equation x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = f (t,x (t) ,x(t− τ(t))) + c(t)x′(t− τ(t)). (1.1) where p, q, r are continuous real-valued functions. The function c : R −→ R+ is continuously differen- tiable, τ : R −→ R+ is twice continuously differentiable and f : R×R×R −→ R is continuous in their respective arguments. To show the existence of positive periodic solutions, we transform (1.1) into an integral equation and then use Krasnoselskii’s fixed point theorem. The obtained integral equation splits in the sum of two mappings, one is a contraction and the other is compact. In this paper, we give the assumptions as follows that will be used in the main results. (h1) There exist differentiable positive T-periodic functions a1 and a2 and a positive real constant ρ such that   a1(t) + ρ = p(t), a′1 (t) + a2 (t) + ρa1(t) = q (t) , a′2 (t) + ρa2(t) = r (t) . Received 14th August, 2016; accepted 6th October, 2016; published 1st March, 2017. 2010 Mathematics Subject Classification. 34K13, 34A34, 34K30, 34L30. Key words and phrases. fixed point; positive periodic solutions; third-order delay differential equations. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 136 A THIRD-ORDER DELAY DIFFERENTIAL EQUATION 137 (h2) p,q,r,c ∈ C (R,R+) are T-periodic functions with τ (t) ≥ τ∗ > 0, τ′ (t) 6= 1 for all t ∈ [0,T] and ∫ T 0 p(s)ds > ρ, ∫ T 0 q(s)ds > 0. (h3) The function f(t,x,y) is continuous T-periodic in t and continuous in x and y. The organization of this paper is as follows. In section 2, we introduce some notations and lemmas, and state some preliminary results needed in later section, then we give the Green’s function of (1.1), which plays an important role in this paper. In section 3, we present our main results on existence of positive periodic solutions of (1.1). Lastly in this section, we state Krasnoselskii’s fixed point theorem which enables us to prove the existence of positive periodic solutions to (1.1). For its proof we refer the reader to [23]. Theorem 1.1 (Krasnoselskii). Let M be a closed convex nonempty subset of a Banach space (B,‖.‖). Suppose that H1 and H2 map M into B such that (i) x,y ∈ M, implies H1x + H2y ∈ M, (ii) H1 is compact and continuous, (iii) H2 is a contraction mapping. Then there exists z ∈ M with z = H1z + H2z. 2. Green’s function of third-order differential equation For T > 0, let PT be the set of all continuous scalar functions x, periodic in t of period T . Then (PT ,‖.‖) is a Banach space with the supremum norm ‖x‖ = sup t∈R |x(t)| = sup t∈[0,T ] |x(t)| . We consider x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = h (t) , (2.1) where h is a continuous T-periodic function. Obviously, by the condition (h1), (2.1) is transformed into { y′(t) + ρy(t) = h(t), x′′(t) + a1(t)x ′(t) + a2(t)x(t) = y(t). Lemma 2.1 ( [3]). If y,h ∈ PT , then y is a solution of equation y′(t) + ρy(t) = h(t), if only if y(t) = ∫ t+T t G1(t,s)h(s)ds, (2.2) where G1(t,s) = exp (ρ (s− t)) exp (ρT) − 1 . (2.3) Corollary 2.1. Green function G1 satisfies the following properties G1(t + T,s + T) = G1(t,s), G1(t,t + T) = G1(t,t) exp (ρT) , G1 (t + T,s) = G1(t,s) exp (−ρT) , G1(t,s + T) = G1(t,s) exp (ρT) , ∂ ∂t G1(t,s) = −ρG1(t,s), ∂ ∂s G1(t,s) = ρG1(t,s), and m1 ≤ G1(t,s) ≤ M1, where m1 = 1 exp (ρT) − 1 , M1 = exp (ρT) exp (ρT) − 1 . 138 NOUIOUA, ARDJOUNI, MERZOUGUI AND DJOUDI Lemma 2.2 ( [21]). Suppose that (h1) and (h2) hold and R1 [ exp (∫ T 0 a1(v)dv ) − 1 ] Q1T ≥ 1, (2.4) where R1 = max t∈[0,T ] ∣∣∣∣∣∣ ∫ t+T t exp (∫ T 0 a1(v)dv ) exp (∫ T 0 a1(v)dv ) − 1 a2 (s) ds ∣∣∣∣∣∣ , Q1 = ( 1 + exp (∫ T 0 a1(v)dv ))2 R21. Then there are continuous T -periodic functions a and b such that b(t) > 0, ∫ T 0 a(v)dv > 0, and a(t) + b(t) = a1(t), b ′(t) + a(t)b(t) = a2(t), for t ∈ R. Lemma 2.3 ( [25]). Suppose the conditions of Lemma 2.2 hold and y ∈ PT . Then the equation x′′(t) + a1(t)x ′(t) + a2(t)x(t) = y(t), has a T periodic solution. Moreover, the periodic solution can be expressed by x(t) = ∫ t+T t G2(t,s)y(s)ds, (2.5) where G2(t,s) = ∫ s t exp [∫ v t b(u)du + ∫ s v a(u)du ] dv + ∫ t+T s exp [∫ v t b(u)du + ∫ s+T v a(u)du ] dv[ exp (∫ T 0 a(v)dv ) − 1 ][ exp (∫ T 0 b(v)dv ) − 1 ] . (2.6) Corollary 2.2. Green’s function G2 satisfies the following proprieties G2(t + T,s + T) = G2(t,s), G2(t,t + T) = G2(t,t), G2(t + T,s) = exp ( − ∫ T 0 b(v)dv )[ G2 (t,s) + ∫ t+T t E (t,u) F (u,s) du ] , ∂ ∂t G2(t,s) = −b(t)G2(t,s) + F (t,s) , ∂ ∂s G2(t,s) = a(t)G2(t,s) −E (t,s) , where E (t,s) = exp (∫ s t b(v)dv ) exp (∫ T 0 b(v)dv ) − 1 , F (t,s) = exp (∫ s t a (v) dv ) exp (∫ T 0 a (v) dv ) − 1 . Lemma 2.4 ( [21]). Let A = ∫ T 0 a1(v)dv and B = T 2 exp ( 1 T ∫ T 0 ln (a2(v)) dv ) . If A2 ≥ 4B, (2.7) then min {∫ T 0 a(v)dv, ∫ T 0 b(v)dv } ≥ 1 2 ( A− √ A2 − 4B ) = l, max {∫ T 0 a(v)dv, ∫ T 0 b(v)dv } ≤ 1 2 ( A + √ A2 − 4B ) = L. A THIRD-ORDER DELAY DIFFERENTIAL EQUATION 139 Corollary 2.3. Functions G2, E and F satisfy m2 ≤ G2(t,s) ≤ M2, E (t,s) ≤ eL el − 1 , F (t,s) ≤ eL, where m2 = T (exp (L) − 1)2 , M2 = T exp (∫ T 0 a1 (v) dv ) (exp (l) − 1)2 . Lemma 2.5 ( [11]). Suppose the conditions of Lemma 2.2 hold and h ∈ PT . Then the equation x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = h (t) , has a T -periodic solution. Moreover, the periodic solution can be expressed by x(t) = ∫ t+T t G(t,s)h(s)ds, (2.8) where G (t,s) = ∫ t+T t G2 (t,σ) G1 (σ,s) dσ. (2.9) Corollary 2.4. Green’s function G satisfies the following properties G(t + T,s + T) = G(t,s), G (t,t + T) = G (t,t) exp (ρT) , ∂ ∂t G(t,s) = (exp (−ρT) − 1) G1 (t,t) G2 (t,s) − b (t) G (t,s) + ∫ t+T t F(t,σ)G1 (σ,s) dσ, ∂ ∂s G(t,s) = ρG (t,s) , and m ≤ G(t,s) ≤ M, where m = T2 (exp (l) − 1)2 (exp (ρT) − 1) , M = T2 exp ( ρT + ∫ T 0 a (v) dv ) (exp (l) − 1)2 (exp (ρT) − 1) . 3. Main Results In this section we will study the existence of positive periodic solutions of (1.1). Lemma 3.1. Suppose (h1)-(h3) and (2.4) hold. The function x ∈ PT is a solution of (1.1) if and only if x (t) = Z (t) (exp (ρT) − 1) G (t,t) x (t− τ (t)) + ∫ t+T t G (t,s){f (s,x (s) ,x (s− τ (s))) −R (s) x (s− τ (s))}ds, (3.1) where R (s) = (c′ (s) + c (s) ρ) (1 − τ′ (s)) + c (s) τ′′ (s) (1 − τ′ (s))2 , (3.2) Z (t) = c (t) 1 − τ′ (t) . (3.3) Proof. Let x ∈ PT be a solution of (1.1). From Lemma 2.5, we have x (t) = ∫ t+T t G (t,s) [f (s,x (s) ,x (s− τ (s))) + c (s) x′ (s− τ (s))] ds = ∫ t+T t G (t,s) f (s,x (s) ,x (s− τ (s))) ds + ∫ t+T t G (t,s) c (s) x′ (s− τ (s)) ds. (3.4) 140 NOUIOUA, ARDJOUNI, MERZOUGUI AND DJOUDI Performing an integration by parts, we get∫ t+T t G (t,s) c (s) x′ (s− τ (s)) ds = ∫ t+T t c (s) (1 − τ′ (s)) x′ (s− τ (s)) 1 − τ′ (s) G (t,s) ds = ∫ t+T t c (s) 1 − τ′ (s) G (t,s) dx (s− τ (s)) = c (s) 1 − τ′ (s) G (t,s) x (s− τ (s)) ∣∣∣∣t+T t − ∫ t+T t ∂ ∂s [ c (s) 1 − τ′ (s) G (t,s) ] x (s− τ (s)) ds = Z (t) (exp (ρT) − 1) x (t− τ (t)) G (t,t) − ∫ t+T t R (s) G (t,s) x (s− τ (s)) ds, (3.5) where R and Z are given by (3.2) and (3.3), respectively. We obtain (3.1) by substituting (3.5) in (3.4). Since each step is reversible, the converse follows easily. This completes the proof. � Define the mapping H : PT → PT by (Hϕ) (t) = ∫ t+T t G (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −R (s) ϕ (s− τ (s))}ds + Z (t) (exp (ρT) − 1) G (t,t) ϕ (t− τ (t)) . (3.6) Note that to apply Krasnoselskii’s fixed point theorem we need to construct two mappings, one is a contraction and the other is compact. Therefore, we express (3.6) as (Hϕ) (t) = (H1ϕ) (t) + (H2ϕ) (t) . where H1,H2 : PT → PT are given by (H1ϕ) (t) = ∫ t+T t G (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −R (s) ϕ (s− τ (s))}ds, (3.7) and (H2ϕ) (t) = Z (t) (exp (ρT) − 1) G (t,t) ϕ (t− τ (t)) . (3.8) To simplify notations, we introduce the constants α = max t∈[0,T ] |Z (t)| , β = max t∈[0,T ] {b(t)} , δ = exp (L) exp (l) − 1 , γ = exp (ρT) − 1. (3.9) In this section we obtain the existence of a positive periodic solution of (1.1) by considering the two cases; c (t) ≥ 0 and c (t) ≤ 0 for all t ∈ R. For a non-negative constant K and a positive constant L we define the set D ={ϕ ∈ PT : K ≤ ϕ ≤ L} , which is a closed convex and bounded subset of the Banach space PT . In case c (t) ≥ 0, we assume that there exist a positive constant η such that η ≤ Z (t) , for all t ∈ [0,T] , (3.10) αMγ < 1, (3.11) and for all s ∈ [0,T] , x,y ∈ D K (1 −ηmγ) mT ≤ f (s,x,y) −R (s) y ≤ L (1 −αMγ) MT . (3.12) Lemma 3.2. Suppose (h1)-(h3), (2.4), (2.7) and (3.10)-(3.12) hold. Then H1 : D → PT is compact. A THIRD-ORDER DELAY DIFFERENTIAL EQUATION 141 Proof. Let H1 be defined by (3.7). Obviously, H1ϕ is continuous and it is easy to show that (H1ϕ) (t + T) = (H1ϕ) (t). For t ∈ [0,T] and for ϕ ∈ D, we have |(H1ϕ) (t)| = ∣∣∣∣∣ ∫ t+T t G (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −R (s) ϕ (s− τ (s))}ds ∣∣∣∣∣ ≤ MT L (1 −αMγ) MT = L (1 −αMγ) . Thus from the estimation of |(H1ϕ) (t)| we have ‖H1ϕ‖≤ L (1 −αMγ) . This shows that H1 (D) is uniformly bounded. To show that H1 (D) is equicontinuous, let ϕn ∈ D, where n is a positive integer. Next we calculate d dt (H1ϕn) (t) and show that it is uniformly bounded. By using (h1), (h2) and (h3) we obtain by taking the derivative in (3.7) that d dt (H1ϕn) (t) = ∫ t+T t [ (exp (−ρT) − 1) G1 (t,t) G2 (t,s) − b (t) G (t,s) + ∫ t+T t F(t,σ)G1 (σ,s) dσ ] × [f (s,ϕn (s) ,ϕn (s− τ (s))) −R (s) ϕn (s− τ (s))] ds. Consequently, by invoking (3.9) and (3.12), we obtain∣∣∣∣ ddt (H1ϕn) (t) ∣∣∣∣ ≤ [(1 − exp (−ρT)) M1M2 + Mβ + M1δT] L (1 −αMγ)M ≤ D, for some positive constant D. Hence the sequence (H1ϕn) is equicontinuous. The Ascoli-Arzela theorem implies that a subsequence (H1ϕnk ) of (H1ϕn) converges uniformly to a continuous T-periodic function. Thus H1 is continuous and H1 (D) is contained in a compact subset of D. � Lemma 3.3. Suppose that (3.11) holds. If H2 is given by (3.8), then H2 : D → PT is a contraction. Proof. Let H2 be defined by (3.8). It is easy to show that (H2ϕ) (t + T) = (H2ϕ) (t). Let ϕ,ψ ∈ D, we have ‖H2ϕ−H2ψ‖ = sup t∈[0,T ] |(H2ϕ) (t) − (H2ψ) (t)| ≤ αγM ‖ϕ−ψ‖ . Hence H2 : D → PT is a contraction by (3.11). � Theorem 3.1. Suppose that conditions (h1)-(h3), (2.4), (2.7) and (3.10)-(3.12) hold. Then equation (1.1) has a positive T -periodic solution x in the subset D. Proof. By Lemma 3.2, the operator H1 : D → PT is compact and continuous. Also, from Lemma 3.3, the operator H2 : D → PT is a contraction. Moreover, if ϕ,ψ ∈ D, we see that (H2ψ) (t) + (H1ϕ) (t) = γZ (t) G (t,t) ϕ (t− τ (t)) + ∫ t+T t G (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −R (s) ϕ (s− τ (s))}ds ≤ γαML + L (1 −αMγ) = L. On the other hand (H2ψ) (t) + (H1ϕ) (t) = γZ (t) G (t,t) ϕ (t− τ (t)) + ∫ t+T t G (t,s){f (s,ϕ (s) ,ϕ (s− τ (s))) −R (s) ϕ (s− τ (s))}ds ≥ γαmK + K (1 −αmγ) = K. This shows that H2ψ +H1ϕ ∈ D. Clearly, all the Hypotheses of Theorem 1.1, are satisfied. Thus there exists a fixed point x ∈ D such that x = H1ψ + H2ϕ. By Lemma 3.1 this fixed point is a solution of (1.1) and the proof is complete. � 142 NOUIOUA, ARDJOUNI, MERZOUGUI AND DJOUDI In the case c (t) ≤ 0, we substitute conditions (3.10)-(3.12) with the following conditions respectively. 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Ge, Periodic solutions for a second order nonlinear functional differential equation, Applied Mathematics Letters, 20 (2007), 110-115. [26] W. Zeng, Almost periodic solutions for nonlinear Duffing equations, Acta Math. Sinica (N.S.) 13 (1997), 373-380. [27] G. Zhang, S. Cheng, Positive periodic solutions of non autonomous functional differential equations depending on a parameter, Abstr. Appl. Anal. 7 (2002) 279–286. 1Faculty Mathematics and Informatics, Department of Mathematics, Univ M’sila, P.O. Box 166, 28000 M’sila, Algeria 2Faculty of Sciences and Technology, Department of Mathematics and Informatics, Univ Souk Ahras, P.O. Box 1553, Souk Ahras, 41000, Algeria 3Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics, Univ Annaba, P.O. Box 12, Annaba 23000, Algeria ∗Corresponding author: abd ardjouni@yahoo.fr 1. Introduction 2. Green's function of third-order differential equation 3. Main Results References